Variable electrical resistor



May 6, 1958 n. KATZ VARIABLE ELECTRICAL RESISTOR 5 Sheets-Sheet 1 Filed Dec. 15. 1955 IHH INVENTOR y 1958 D. KATZ 2,833,901

VARIABLE ELECTRICAL RESISTOR Filed Dec. 15, 1955 5 Sheets-Sheet 2 FIG] / llllll- INVENTOR D. KATZ VARIABLE ELECTRICAL RESISTOR May 6, 1958 5 Sheets-Sheet 3 FIG. l8.

Filed Dec. 15, 1955 INVENTOR y 6, 1958 D. KATZ 2,833,901

VARIABLE ELECTRICAL RESISTOR Filed Dec. 15, 1955 5 Sheets-Sheet 4 1 H H },I

FIG. I40

1N VENTOR y 6, 1958 n. KATZ 2,833,901 7 VARIABLE ELECTRICAL RESISTOR Filed Dec. 15, 1955 5 Sheets-Sheet 5 FIG. l5. FIG. I50.

y km 7 I IN VENTOR United States Patent O VARIABLE ELECTRICAL RESISTOR David Katz, Wilmington, Del.

Application December 15, 1955, Serial No. 553,334

15 Claims. (Cl. 201-'-62) This application is a continuation-in-part of my application Serial No. 518,240, filed June 27, 1955.

The invention herein deals with variable printed resistors. It is an object of this invention to provide variable resistors, adapted to be manufactured by recently developed printing techniques for electric circuits. Another object is to provide variable resistors which can be designed to vary resistance, in response to uniform (linear or circular) motion of a sliding contact member, according to any preassigned mathematical formula which is consistent with the natural properties of resistance. A still further object of this invention is to provide variable resistors which are compact in bulk and economical to manufacture. Other objects and achievements of my invention will become apparent as the description proceeds.

In a general way, the objects of my invention are accomplished by printing a series of closely spaced, essentially uniform, parallel (or radiating) lines with a conducting ink on a non-conducting surface, for instance a plastic sheet or ceramic plate. Then one edge of the imprinted surface is trimmed into an essentially straight (or circular) line cutting across the imprinted lines and being adapted for contact with a sliding contact member,

while the opposite portion of the sheet is contacted with a fixed conducting member whose edge is shaped to form a smooth curve bearing a predetermined mathematical relation to said first edge. Terminal wires (leads) or other conducting elements, aifixed to said sliding contact member and to said fixed, curved conducting member, complete the assembly. When the lead wires are connected to a source of electrical energy, current flows through said sliding contact member, and through whatever imprinted lines are in contact therewith, to said fixed, curved conductor, and out through its terminal or other conducting element.

For better understanding of my invention, reference is made to the accompanying drawings, wherein Fig. 1 is a plan view of one form of my invention.

Fig. 2 is a section through Fig. 1, along line 22.

Figs. 3, 4 and 5 are perspective views of three different sliding contactors that can be used with the resistor shown in Fig. 1.

Fig. 6 is a plan view of a modification of my invention wherein two difierent resistors are contacted in two different manners by the same slide.

Fig. 7 is a perspective view of the sliding contactor employed in Fig. 6.

Fig. 8 is an exploded perspective view of a cylindrical modification of my invention, employing a revolving contactor.

Fig. 9 is a plan view of a still different modification of my invention, wherein the resistor is made up of lines radiating from a center and wherein a revolving contactor is employed in the center of the resistor plate.

Fig. 10 is a difierent modification of a radiating resistor, wherein the revolving contactor is on the rim of the resistor plate.

Fig. 11 is an exploded perspective view of an improvement upon the basic form of my invention shown in the preceding figures.

Fig. 12 is a plan view of another modification achiev- .ing the same improvement.

2,833,901 Patented May 6, 1958 ice Figs. 13, 14, 15 and 16 are plan views of resistor plates in which the imprinted lines are laid oif in accordance with various functions y=f(X), X being the horizontal lower edge of the resistor plate.

Figs. 13a, 14a, 15a and 16a are mathematical diagrams indicating the various forms of resistor variation obtainable with the resistors of Figs. 13, 14, 15 and 16 respectively, using the three principal types of sliding contact available in this invention.

Figs. 17 and 18 are similar plan views showing resistors wherein the function y=f(X) consists of several sections of different mathematical relations with respect to X, while Fig. 17a is a mathematical graph of the resistance variation obtainable in Fig. 17.

Referring now in detail to Figs. 1 and 2, a plate 11 of dielectric material, is imprinted with a series of parallel, vertical lines 111 by means of a conducting ink, for instance an aluminum, copper or carbon-black ink. The plate 11 may be of ceramic material, and the conducting ink may be a silver ink, in which case the plate is first imprinted and then fired, to produce an intimately bonded imprint. Or the conducting lines may be imprinted on a thin sheet of plastic material, and the latter may then be bonded, for instance by adhesive, to a supporting plate 10 made of any convenient material such as cardboard, wood, plastic or metal.

The lines and their spacing in Fig. l are indicated diagrammatically. Actually, the lines are much finer, and their spacing much closer than shown in this figure. In the printing of plastic sheets with copper foil by the etching method, it is not uncommon to make the lines 0.00135 inch thick and 0.005 inch wide. (Modern Plastics, August 1951, pages 99 to 111.) In printing paper with carbon inks, very fine printing may likewise be achieved.

The lower edge of the imprinted sheet 11 is trimmed near a straight line 113, which represents the X-axis, while the upper edge is in contact with a fixed metallic conductor 12 whose lower edge is shaped according to a designated mathematical function y=f(X). In Fig. '1 and in the upper half of Fig. 6, this function is y=kX wherein X is the distance (in cm.) along said straight edge 113. In the lower half of Fig. 6, y is a linear function of X.

The portion of the imprinted sheet beyond the conductor 12 may be cut away, as in Fig. 1, or allowed to remain, as in Figs. 8 and 11. Conductor 12 is provided with a terminal or lead-wire 123, located at any convenient position along its length.

In contact with said lower edge 113 is a sliding conductor 16 or 15, which can be moved continuously along said lower edge. Slide 16, is shown in Fig. 1 and is detailed in Fig. 3. It is designed to make contact with the entire group of imprinted lines 111 which lie between edge 104 of plate 11 and the forward end 162 of slide 16. Or, it may be slid in from the opposite direction, in which event, the group of lines contacted would be that lying between edge 105 of plate 11 and end 162 of slide 16. A terminal 163, at a convenient location, completes the electrical structure of slide 16. This slide will hereinafter be referred to as the ribbon-type of slide.

Slide 15 is of the spur-type or roller-type, and is shown in Figs. 4 and 5. The contact element is a spur 151 or roller 155 (mounted on a pin 156). The principal stem 15 is provided at one end (or at some other convenient location) with terminal 153. Contact by spur 151 is made with a limited bundle of imprinted lines, whose location on sheet 11 will obviously depend on the position of spur 151 (or roller 155) along the X-axis.

It is clear that when terminals 123 and 163 (or 153) a are connected to a source of E. M. F., current will flow through the body of the slide across edge 161 (or through spur '151), into the group of imprinted lines 111 in contact with the latter, and then through the body of fixed conductor12, out through terminal 123.

With a printed plate of the above type, contact may be made with a portion of the imprinted line along one edge of the carrier plate. For this purpose, each y on sheet 11 is laid off with small extensions z and Z; on the ends.

'Portion z may be in the same plane with portion y as shown in Figs. 1, 6 and 8, or it may be bent over the edge of supporting plate 10, as in Figs. 11 and 12. In the former event, slide 16 is given a cross-sectional shape as shown in Fig. 3, whereby it makes contact on the surface of plate 11 along the extensions Z1. Along the upper edge of sheet 11, the fixed contact member 12 is likewise shaped to cover up the extensions Z2 of the ordinates. Obviously, the mathematical relation y=f(X) above postulated, contemplates as y the length of each ordinate between said extensions (exclusive of the latter).

In Figs. 11 and 12, a portion 112 of the imprinted sheet 11 hangs over the front edge of supporting plate 10, and is pressed against said edge by the slide 16 (or 15). This overhanging portion corresponds to the extension z in the other modifications.

The imprinted y-lines need not be perpendicular to the x-axis. Instead, they may be inclined, as shown in Fig. 6, provided it is remembered that it is the length of each y (exclusive of its extensions 2), and not its vertical projection, that should correspond to the basic formulas above derived. Of course, where the conducting lines are imprinted on a thin, plastic sheet or paper, it

is not necessary to print the lines in inclined form. Instead, the lines may be printed perpendicular to the edge of the sheet, but the latter may then be cut out properly to give the sloping lines indicated in Fig. 6.

In a sheet with inclined lines, the forward edge 162 of the slide 16 (or 152 of slide 15) is cut off at an angle 0 corresponding to the inclination of the imprinted lines. In other words, the forward edge of the sliding contactor is always parallel to the lines in contact therewith. This detail is shown in Fig. 7, wherein the sliding member is of the area-type on one side and of the spur-type on the other side.

Conductor 12, as well as sliding contactors 16 and 15, may be made of any convenient conducting material, for instance, copper, silver, aluminum or electrode carbon. Conductor 12 may also consist simply of a continuous strip or ribbon imprinted by means of a conducting ink across the upper ends of lines 111, provided, as already mentioned, that the lower edge 121 of this strip or ribbon conforms to the curve y=f(X) prescribed by the particular problem on hand.

Mathematical theory Referring primarily to Fig. 1, let X represent the abscissa of any point on curve 121, in cm., and let y represent its ordinate.

where AA represents the cross section of one of the imprinted lines. For instance, if lines of copper foil of the above mentioned width and thickness are imprinted,

AA=0.005 0.0013S, i. e.. 6.75 sq. in. orabout 43.5 10 sq. cm. The conductor being copper, p is about 1.7x l0- The resistance of each element, then is Then the resistance of about 0.039 If y is given a height of cm. at its highest point, the resistance of the highest line is about 1 ohm. This design is suitable for use with a spur-type contactor as shown in Fig. 3. I

For use with a ribbon-type slide (as typified by member 16 in Fig. 1), imprinted lines of higher resistance are desirable. For this purpose therefore a carbon or graphite ink is to be chosen. Depending on the concentration of the carbon in its dielectric vehicle, .the resistance of the imprinted lines may be varied from a few ohms to 200 megohms per inch of length. (Printed Circuit Techniques, National Bureau of Standards Circular 468, page 15. The -value will vary accordingly, An element of width b then has a conductance of v p Now, the conductance of the group of lines 111 contacted by slide 16 when its forward end 162 is in position X, is the integral sum of the elemental conductances between this point X and the terminal abscissa L.

L L G=f dG=fl Q 2 X b X y For instance, assuming the copper foil above to be etched to the dimensions above indicated; spacing the lines 0.010 inch apart, b=O.0l5 inch=0.038 cm. Then Assuming that we shaped edge 121 to represent y=kX', the above integral gives us Assuming L=25 cm.,

X is a variable determined by the position'of slide 16, while k is a parameter which we can vary at will to achieve whatever purpose we find desirable. For instance, if we wish to make the height of the resistor just about equal to its width, we set y =L; kL =L;

Then y =0.04X is the shape needed for edge conductor 12. With this value of k,

Resistance being the reciprocal of conductance,

infinite (open circuit). I

Sometimes it is desirable to express the resistance -R laid off by contactor 16 in terms of x instead of X,

wherein x is the fraction of the total width L-marked off by end 162 of slide 16. Then "when substituted in (2b) above, this gives and assuming L=25 and k=0.04,

When x=0, R=; when x=l, R= when x=%,

R=1.49 milliohms.

Heavier values of R can be obtained by using a sheet printed with a high resistance carbon ink, in lieu of a sheet etched from copper foil.

In the design of Fig. 1, R becomes zero when the slide reaches the origin, because then it comes in direct contact with fixed conductor 12. Therefore, slide 16 cannot be moved in from the left end toward the right. If, however, y is given a positive value above zero at the extreme left end, the motion of the slide can be made to be from left to right. Printed plates of such configuration are shown in Figs. 13, 14, 16, 17 and 18.

When moving in slide 16 from left to right, Formula 2 must be replaced by AA MK 1 07 Replacing, for convenience, X by xL, and

by I (a constant, once the layout of the lines and the composition of the printing ink has been settled upon), the two principal formulas for G become:

Case (a): Slide moves in from left edge:

G=JL (6a) 0 2/ Case (b): Slide moves in from right edge:

a ll

By way of illustration, several typical forms of y are indicated in Figs. 13 to 16 inclusive. Table 1 below gives the corresponding formulas for R. The general shapes of the R-vs.-x curves obtainable in these cases are given in Figs. 13a to 16a, respectively, which, for completeness, indicate further the R-curves obtainable with a spur-type (or roller-type) contact.

It is to be understood, however, that these curves are meant merely to illustrate the principle and are not drawn to scale. For the same reason, compound constants, such as are replaced in the table by simple constants, such as a, c, k and h.

TABLE 1 Case I-Uslng spuror roller-type contact.

Case II-Using ribbon-type contact. a-Sliding from x=0 toward x=1 b-Sliding from x=1 toward x=0 The mathematical theory of the surface.

se w.

is similar to that of the spur-type, except that here AX depends on various factors (e. g. the pressure of contact) and is best determined experimentally (as taught below).

Returning now to the formula derived above for the resistor indicated in Fig. 1, it will be noted that when k and L are selected so that kL=l, the actual magnitude of L does not enter into the calculations for the resistance. L, however, is material in determining the fineness with which the resistor may be adjusted and read. The longer L, the more of lines 111 can be printed thereon. Then the change produced by moving slide 16 a distance I) will constitute a finer fraction of the total resistance laid off by the slide.

Miscellaneous modifications Although fiat supporting plates 10 have been discussed up to now, it is clear that in most cases the imprinted sheet 11 can be wrapped around a cylindrical In such event, the imprinted lines may be disposed at right angles to the axis of the cylinder whereupon the slides 16 or 15 move in a straight line parallel to the axis; or the imprinted lines may be disposed parallel to the axis, in which event a spur-type or roller type, revolving contactor may be employed along the rim of the cylinder.

Such a modification is shown in Fig. 8, wherein the support 10 is of semicylindrical form, and the front portion of it is made integral with a non-conducting block 101 which carries a conducting pin 102. The latter is adapted to receive revolving contactor 15 (of the roller type) and the dielectric knob 157 which is notched at 158 to fit over the shank of contactor 15. The leadwire 153 may be connected to pin 102 at any convenient point thereon, 103.

A scale 116 on the front face of semicylinder 10 completes the assembly. Similar graduated scales may be provided with any of the other modifications of my invention discussed hereinabove or hereinbelow, to indicate the amount of displacement (linear or circular) of the sliding contactor. Such scales may he graduated in centimeters or other fixed units, or they may be designed to indicate the position of the slide as a fraction of the total possible displacement.

Still diiferent modifications of my novel resistor are indicated in Figs. 9 and 10, which are best suited for use with a spur-type (or roller-type) contact. In these two figures, the parallel lines 111 of Fig. l are replaced by lines radiating from a center. In Fig. 9, the center of the supporting plate is hollow and admits a revolving contact member 15, having a contacting spur 151 and the customary lead-wire 153. The member 15 is mounted on a pivot in the center of the hollow circle, and contacts the imprinted lines along circular zone Z1. The outer regions of the imprinted lines are in fixed contact with conductor 12, which corresponds to conductor 12 in Fig. 1, and has any desirable shape corresponding to a selected function p=f(0). For instance, we may use the form of Fig. 8 for =a0, =a0 =a /0 or 2:11 log (1+0), a being a constant. Electric terminals 123 and 153 are provided on the fixed and rotating contact members, respectively, as in the other modifications.

In Fig. 10, the revolving spur-type contactor is on the outer rim 112 of the plate; the rim then, obviously is circular. The imprinted lines 111 then radiate from this rim inwards, toward an optionally selected center. The region around the center, however is cut away and rimmed by a fixed conductor 12 having any desirable matheroller-type contactor matical shape (e. g., a circle or an ellipse).

The modification of Fig. 10 is suited for such problems as V Of course, the modifications shown in Figs. 9 and 10 need not embrace a full circle; semicircular resistors, quadrant resistors, or resistors covering any other fraction of a circle may be built on the same principles.

There remains now the problem of wear. With the ribbon-type and spur-type contactors, continued use is apt to rub oif part of the imprinted ink, thereby increasing the contact resistance of the system. In the case of the spur-type contactor, the problem may be solved by resorting to a roller-type contact, as illustrated in Figs. 5 and 8. For the ribbon-type contact, however, the special modifications of Figs. ll and 12 come into consideration.

In Fig. 11, the fiat support 10 has one edge thereof 102 finished off to a smooth surface, as by planing and polishing, and sheet 11 with its rulings are bent over to form flap 112, which rests against the polished face 102. If the sliding contactor 16 be held flat against this flap, pressing it against face 102, electrical contact will be achieved between lead-wires 163 and 123, through the bodies of conductors 16 and 12 and those of the rulings c 111 whose ends on flap 112 are in contact with slide 16.

After the device has been assembled in the above fashion, the exposed horizontal portion of the rulings (but not the portion on face 112), as well as conductor 12, may be sprayed or brushed with an insulating, weather-protective lacquer or varnish. Slide 16 may likewise be protected or insulated on its exposed surfaces, but not on surface 161 which makes contact with flap 112.

However, as already said, in an assembly ofthis sort there is danger that repeated sliding of contactor 16 over flap 112 may rub off entirely or partially the lines 111, thereby increasing their resistance at the point of contact. Accordingly, I interpose between 16 and 112 a fixed, laminated, commutator 13. The latter is an oblong, laminated block with a smooth face 132 adapted to rest against flap 112 and another smooth face 133 adapted to make contact with slide 16. The laminations are made up of thin metallic foil insulated from each other as by a coating of varnish or thermosetting resinous material. The individual laminae extend at right angles to the length of the block, and their individual thickness is not greater than the width of each ruling 111; preferably, it is several times thinner. In any event, the aim is to achieve harmony between the period of the laminations and the period of the rulings 111. (By period, I mean the distance between the face of one lamina or edge of one line and the corresponding face or edge of the adjacent lamina or line).

Commutator 13 may be assembled from its lamina in any convenient manner, and then pressed together, with heating if necessary, to make the assembly coalesce into a single, rigid body. Paces 132 and 133 may then be prepared on it as by planing, cutting or polishing. If

desired, an oblong metallic bolt 134, suitably coated with an insulating varnish, may be thrust through the pile to provide means for holding it tightly together and for locking it tightly against fiap 112 by means of cooperating locking members on member 13 and support 10, for

instance hooks 135 and pins 103. v I

The commutator 13 is thus permanently fixed against flap 112 and provides channelledelectric contact between rulings 111 on the latter and slide 16. The latter is mounted to slide over surface 133 with considerable pressure, inasmuch as the danger of wearing out the contact is now absent. The requisite pressure can be achieved by aid of rubber rollers 181 held on plate 18,

, which may befastened to the. underside of block 10 by means of screw holes 182. All exposed parts of member 13 (but not its faces 132 and 133) may be sprayed or brushed with an insulating coating.

, ductance continually increases.

wherever possible.

In Fig. 12 a difierent mode of achieving the same result is shown. Here a soft, flexible conducting ribbon 14 is interposed between overhanging flap 112 and face 161 of the sliding contactor 16. This ribbon may be a thin ribbon of metal, or it may consist of a woven metallic ribbon or indeed-a textile ribbon having interwoven metallicthreads therein. One end 141 of ribbon 14 is fastened to edge 104 of block 10, while the other end passes over the end roller of the group 181, then over additional rollers if need be, and is fastened to the final roller 183, which is mounted on pin 184 carried by an extension of member 18. A spiral spring (not shown) affixed to said final roller pin, holds ribbon 14 under tension, and consequently out of contact with face 112 But as slide 16 moves in, it presses a portion of said ribbon against the rulings on flap 112; A notch or mark 164 on the .body of'slide 16 serves to indicate the point at which ribbon 14 passes from contact to no-contact with flap 112.

Principles of design The mathematical theory above teaches how to determine the variation of R with X when the shape y= f(X) of the lower edge of member 12 is given. In practice, however, the problem is often reversed. The practical application contemplated may dictate the mode of variation of the resistance R with the position X of the sliding cont-actor, and the problem then is to determine the requisite shape y f(X) of said lower edge. The solution to this problem is readily achieved from the mathematical principles above established. Thus, if a spur (or roller) contact is to be used, Formula 1 gives the basic solution.

If we let x=X/L and AA n Formua 1 becomes simplified into y wherein AX is the width of the spur (or the estimated width of contact between the roller and the printed sheet). .From this we obtain wherein K=JAX=a constant. Thus y is proportional to R, and the curve y=f(X) has the same general shape as the assigned curve for R. The spur (or roller) contact,

is thus suitable for any assigned form of variation of R ances (short circuit) are admissible at selected points, but

R can have no negative values. p 7

With the ribbon-type of slide, the choice of R is more limited. First of all, since the slide of type 16 contacts more and more printed lines as it slides along, the con- R, therefore, must be a continually decreasing function in the direction of the slide motion. Secondly, as before, R must have positive values only. A zero value is admissible only at the very end of the slide displacement, inasmuch as once such a point is contacted, the circuit will stay shorted for the rest of the path of the slide. Thirdly, since the conduct-- ance G is expressed by a definite integral, it always has the form of a difference between two algebraic expressions: If the slide moves from the origin of X forward [hereinafter" designated as J[F(X)F(O)l, integral 1 case (0)], this expression is wherein F (X) represents the indefinite 9 10 opposite end toward the origin [case (b)], then G has For more general treatment, however, it is best to trans the form J [F(L)F(X )1. It follows that R must have form the assigned form of R into one or the other of the one or the other of the forms: following forms:

J[F(X) F(0)] and J[F(L) F(X)] R= 1n case (a) (Slide moves from To satisfy these forms by an assigned curve R=(X), it x=0 forward) is essential that (X) become infinite at X =0 for case and (12) (a), and at X=L for case (b). 9

The analysis from this point on becomes simpler if we in case (Slide moves from replace X by xL; then R becomes a function of x, which x: 1 toward x=0) represents the position of the end 162 as a fraction of the total length L of the X-axis. Then, G must have the 'form 0 may be a constant or a funcnon of x whlch does a d ma-m1 (a) salts-teale 53? W) n JL[f(1) f(a;)] in case (b) livnmlig flor simplicity t in lieu of 0(0), 1/1 in lieu of correspondingly, R must have the forms M leu o 1 a Firm (a) dx and (10) and 0 in lieu of in case (b) do [f )f-' '2; f(x) above symbolizes the lniliifinlte integral the Solution for y in such event is:

J L6 Then y=mt in case (a) 1 and 13) ,incase (b) and g 0w+(1 1 To illustrate, if we want fw) wherein 2:v i- 5 d 1-:0

then0=2x+5,==x,0 =2, =1,\,0 =1. Substitutionof Therefore, if the required R is assigned in the form these Valuesin (13) Yields R=(x), it is merely necessary to convert this form into one or the other of forms (10), which operation yields f(x). Then We determine y by Formula 11.

Such conversion may often be achieved by inspection, 40 If the assigned form of R does not become infinite at as illustrated in Table 2 below. x==0 or at x=1, then the assignment is inconsistent with TABLE 2 [Wanted iorm R (001. 1); rewrite in form R1 (col. 2); then find form of Its) in col. 3, and required form of 1/ in col. 4. Last column indicates type of slide motion required for the particular problem and solution] Ex. R R f(a:) y im 1 1 am JL 1 a...

aa-1-a:c JL a must (b) aO-fl) a-i -w JL 2 3 a a L sin Ljza sec 11 1i 1r IE1! JLa 2 1r 2 1--s1n sin s 2 a a 1 ZIr 4JLa :nr

x b 4 x tanLmngl JLa 4 w 4 U -a a log a: 6 log :v log 1log z JLa (b) az az-MJ JL a 001m- (a) 1 1 3 JL 7 as:

u.'ra-0 s a 'L i 17 2JLa g sin? sin -sin 0 Sm 2 sec 2 (a) 121r a 1 It 4JLa 2 1i- 9 a 0m tan cos 4 tan ii -tan 0 4 4 a a 1 JD: 53:21 75:55 m (a) 1 l the properties and behavior of the ribbon-type slide, and resort must be had to a spur-type or roller-type. slide.

As already indicated, R is always infinite at the beginning of the motion in the ribbon-type slide, and can only decrease from that point on. (In the spur-type slide the resistance can both increase or decrease at optional points.) However, the resistance may be kept constant during a portion of the motion by the simple device of breaking theimprinted lines over a given portion of the X-axis. An illustration of this is presented in Fig. 17, wherein the imprinted lines are broken in the region x to x by simply cutting out a short ribbon from sheet 11 in the corresponding'regions. The resulting R-vs.-ic curve is shownrin Fig. 17a. By resorting to stops of this nature, the value of the resistance R over the range 0 to L (or L to O) can be given the properties of a cam, whereby the current through the circuit controlled by the resistance R can be divided into arbitrary sections, so that the current increases according to any desired law for certain fractions of the period of motion but remains stationary during others. n

In the spur-type slide, similar portions of steady resistance can be introduced simply by making y a horizontal straight line over the corresponding stretches of X. 011 the other hand, resistors having sections of disrupted lines as in Fig. 17 may also be used with a spur-type or roller-type contact. In such event, the resistance value of the entire device becomes infinite (open circuit) as the contactor moves passed the disrupted sections.

Moreover, as a general proposition, the fixed conductor 12 need not be a mathematically continuous curve, but may instead be made up .of a plurality of sections each having its own mathematical relation to the X-axis. An illustration of this is shown in Fig. 18, wherein y equals f (x) from 0 to x f (x), from x to x;; and f (x), from x to l, x being X/L. Such resistors may be used either with the spur (or roller)-type contact, or with the ribbon-type slide, and in the latter case the motion may be from right to left or from left to right.

As a further modification of my invention, the ribbontype slide may be made of a length less than the total stretch of X from 0 to L. In such a case, the basic formula for design becomes G=J[F( )J( wherein B is the horizontal length of the slide, and

regardless from which direction the motion is started,

provided that the entire length B of the slide is Within the region of 0 to L. The rest of the analysis in this I case follows closely the principles above discussed, it being remembered that R=l/G Experimental design procedure such factors as the resistivity of a colloidal suspension 7 of carbon black in a plastic ink-medium, as well as the Width and thickness of an imprinted hair line. case of a roller may involve such difficult problems as how wide is the electrical contact between a metallic AX in the f) cylinder (the roller) and a sheet of thin plastic stretched tight over a block of wood. It is clear that errors in the values of J and AX may upset the accuracy of the resulting design.

To avoid the above difficulty, I propose the following procedure for designing and building the variable resistors Y r of this invention.

Having chosen the requisite plastic film (cellulosic a 12 film, polyvinyl, methacrylate, etc.)., having settled upon the desired values of L, b and AX, having decided upon the ink composition (say, carbon black dispersed to a specified concentration in an urea-formaldehyde polymer), and having estimated the desired width and thickness of the imprinted line, proceed now as follows:

(1) Print the film with the chosen ink, the printed lines being spaced off at intervals corresponding to the chosen b-value.

(2) Cut out a rectangle of the imprinted film, so that one edge thereof is perpendicular to the imprinted lines. Calling this edge the X-axis, let its length be represented by L0.

(3) Apply a straight-line metal conductor of negligible resistance to the lower edge of the rectangle (covering up a region along the X-axis corresponding to z in the above figures). Apply a similar strip to the opposite edge of the rectangle. Let Y represent the height between the inner edges of the two conductors.

(4) Using a dry cell, a millivoltmeter, standard adjustible resistances or potentiometers (or whatever other apparatus may be needed), measure the resistance of the entire square in the direction of the ruled lines. Let T represent this total resistance. Then by Formulas 6a and 3 above,

L I T: 1/G;henee JL Then for any other length L,

Since Y, L L and T are all quantities which can be measured with great accuracy, Expression 15 gives us a highly accurate method for computing IL, for use in all subsequent designs involving the same printing-ink, same width and thickness of imprint, the same spacing between lines, and a ribbon-type slide.

Now, to get a similar expression for the essential constants in the spur-type or roller-type contact, cut out a rectangle of the imprinted sheet as in step (2) above, but apply to it only one straight-edge metal conductor under step (3) above, say at the upper edge of the rectangle. Apply to the lower edge a spur-type contactor as in Fig. 4 or a roller-type contactor as in Fig. 5. Measure the height Y of the imprinted lines between the upper end of the spur or roller and the lower edge of the upper horizontal conductor. Measure the r'esistance T of the bundle of lines thus connected, as in step (4) above. 7 7 w a Then, by Formulas 1 and 3 above,

J AX

hence Y JAX- T (1e) Since Y and T can be accurately measured, their quotient may henceforth be substituted for J AX in all further designs employing the same printing ink, the same weight and pattern of the imprint, and the same spur-slide or roller-slide (or other slides of the material, shape and exact dimensions as the one tested).

It will be clear from all the aforegoing that my invention provides means for building economically variable resistances whose value is a function of the displacement of a sliding member, and whose variation with said displacement can be made to follow practically any desired pattern, thereby fashioning by the aid of such resistances oam-like control devices for numerous mechanical or electrical devices which are susceptible of control by current or voltage.

In a sense, it may be said that my invention brings to the world an electrical cam. a

It will be understood that the details of the design and execution may be varied considerably without departing from the spirit of this invention. For instance, screwfeed devices or micrometer movements may be applied to the slides to increase the accuracy of their setting and reading. Furthermore, where the variable resistor of this invention is used in an automatic control system (for controlling the operation of a machine or of a chemical process), the sliding of the contactors may be achieved mechanically by suitable mechanisms or electrical do I vices installed for the purpose. Proper housing or supports may be provided for the device, and the resistor element may be suitably insulated from such housing or, on the contrary, grounded therethrough. The overhanging flop principle shown in Figs. 11 and 12 may also be used in conjunction with a spur-type or rollertype contact, except that here care is provided to insulate the flap (as by varnishing) or to insulate all parts of the slide member except the face of the spur or roller, so as to prevent electrical contact between the sliding membetand the overhanging flap, except through said spur or roller.

Instead of straight lines, the imprinted lines 111 may be cured, for instance into circular arcs, sinusoidal patterns or any other convenient shape, provided the curves are of a nature that their exact length between the sliding contactor and the fixed conductor 12 can be readily measured or computed. Numerous other variations in detail will be readily apparent tothose skilled in the art.

Summary My invention provides a variable resistor, whose electrical value is determined by the position of a sliding contactor, and which is characterized by the property that the value of the resistance may be varied in response to the displacement of said contactor according to any pre-assigned mathematical function, R=(X). The only limitation upon (X) is that its values in the region employed (X: to X l.) must all be positive. With the special type of sliding contactor which produces a ribbontype contact, certain additional limitations must be taken into consideration, for instance, (a) the resistance must be infinite at the beginning of the slide displacement; (b) the resistance cannot increase in the direction of slide motion; and (c) the resistance must not be zero except at the end of the slide displacement. The spurtype contactor (or its equivalent, the roller-type contactor) is free of these added limitations.

The principal structural features of my novel resistor are embraced in the following characteristics:

1) The resistor is essentially a sheet of dielectric material on which are imprinted, by means of currentconducting ink, mutually separated but closely spaced parallel or radiating essentially uniform lines. printed sheet may be mounted on a flat or curved (cylindrical) supporting block.

(2) The parallel or radiating conducting lines are of various lengths, and are laid out in such a manner as to define with their ends two, smooth, continuous curves. One of these is preferably a straight line or an arc of a circle, while the other one defines a desired mathematical function with reference to said first line as an X-axis or with reference to the center of the radiating lines.

(3) The electrical currents involved in the problem are sent through a bundle of adjacent printed lines, the size of the bundle and its position in the assembly being varied according to the type of contactor selected (ribbon vs. spur) and according to the value of X.

(4) Two principal modes of sending current through the continguously imprinted conductors may be resorted to:

I. The current may be sent through a movable spur (or roller) of limited Width X-wise, as illustrated by Figs. 4 and 5 above.

The a II. The current may be entered through a ribbontype contactor, which touches all the conductors in a region of substantial width along the X-axis. In such a case, the ribbon contactor may embrace all the conductors between a given abscissa X and one edge of the imprinted sheet (i. e. X=0 or X=L). Or it may be designed to contact a group of conductors between any two positions which are a fixed distance apart, for instance abscissas X and (X +B), wherein B is a constant.

In all the cases above, the spur or ribbon contactor is adapted to be slid along an edge of the imprinted sheet to vary the value of X.

(5) The curve defined by the opposite ends of the imprinted conductors is a mathematical curve corresponding to the nature of the assigned problem. The ends of all the conductors along this curve are electrically connected by a fixed electrical conductor running through the full length of said curve.

(6) Electrical terminals along the bodies of said movable contactors on the one hand and of said fixed mathematically shaped conductor on the other hand, provide means for inserting the variable resistor into any desirable electrical circuit.

(7) Means for pressing the moving contactors against the imprinted lines, means for controlling the displacement of said contactors (e. g. screw-feed means), and a scale to indicate the amount of displacement, complete the basic structure.

By essentially uniform lines in the discussion hereinabove or in the claims below are meant lines of uniform width and thickness, wherein the uniformity is as close as can be achieved by ordinary skill with a ruling pen or a printing or engraving machine. In other words the characterization essentially uniform is meant to exclude cases wherein a deliberate effort is made to make the lines wider or thicker in certain sections of their length than in others.

The expression mathematical curve is meant in a broad sense to include cases wherein the permanently connected ends of the imprinted conductors are laid out in sections of which some define separate and distinct functions of X compared to the others. Said expression may also include cases wherein the pattern includes sections of constant y or sections wherein the conducting lines are disrupted so as to present zero conductivity along said particular sections.

The words printed and imprinted used throughout this specification and claims have acquired a broad meaning in the art of printed circuits. In Modern Plastics of August 1951, page 99, cited hereinabove, the usage of the art is expressed in the following words:

Printing is defined as the act of reproducing a design upon a surface by any process. The article then refers to the diverse printed circuit production methods which have been proved in practice, and lists specifically the following:

(1) Printing of silver ink on ceramics followed by high temperature firing to fuse the silver.

(2) Spraying of metal into depressions in a plastic plate.

(3) Stamped metal patterns, either bonded to a plastic base or used as a rigid wiring harness.

(4) Etching of metal foil-clad plastics.

It will be understood therefore that these terms, when used in the claims below, are not meant to be restrictive, but are to be construed in the broad sense indicated by the definition above quoted.

By substantial resistivity in reference to imprinted conductive lines, I mean a definite and measurable resistance per unit length of said lines, as distinguished from the indefinite and negligible resistance of joints, lead wires, distributor heads and switches commonly occurring in electrical circuits.

Lclaimas my invention:

1. Antelcctricalresistor comprisingin combination (1) a sheet-onplate of non-conducting material having imprintedthereon anassemblyof closely adjacent relatively thin, hon-crossing, electrically conductive lines of uniform and substantial resistivity and coveringby their number a substantial 2widthof said sheet or plate comparedtoits averageheight, and (2.) two conductive contact members in contact with said assembly and spacedfrom each other whereby to delimit the electrically effective length of eachimprinted line, the resistance of said cntact members ,beingnegligible comparedto the resistance of a the average imprinted line delimited between them; one. of ,s aid contactmembers being fixed relative to said assemblyand extendingcrosswise of theimprinted'lines whereby ,to' provide a' 'perrnanent conductive connection between the' lines thus contacted at oneen'dof the assemblyaandthe other contactmember being movable across si -a embl a i bein slan d t nt but a s c group pf d'lines'in anyjoneof'its positions; alltothe eif ect that whena potential difference is applied tosaid two contact members, current will flow through a'selected group of saidimprinted lines from one :of the contact members to the o'ther, themagnitude of the current being determinedjby the potential "difi'erenceapplied and-by the summed conductance 'of said selected group of lines which is insimultaneous'contact with both said contact members.

"2. An ,electrical resistor as in, claim :l, saidfixed contact member being shaped'to define acurve of'form y fCX) as the .upperlimitpf said assembly of-lines when the length of each lineis measured'from the-point where it is adapted to be contacted'by saidmovable contact member, X being the abscissa of displacement of said movable contact member from a fixed'origin, and fcX being a pre-assigned function of-X, whereby the-resistance of each line .at the instant of simultaneous contact by both'contact members'is proportional to -y=f(X).

'3. Anelectricalresistor'as in-claim-Z, said moving'contact member being fashionedto contact-the entire group of 'imprinted lines comprisedbetween saidpoint X and one of 'the edges ofsaid-imprintedsheet.

4. An electrical resistor as in claim 1, said moving contact member-"being adapted to contact a-fixed number of said imprinted =l'ines in each position of said contact member.

5. An electricalresistorasin claim -1, the'imprinted linesbeing parallelto each other, and-the movable contact member-beingadapted to move in-a'pathwhichtraces a straight line on-the sheet: containing the-imprinted assembly, said straight-line intersectingsaid par'allel'lines.

=6. An electrical -resister-asin claim 1, the-imprinted lines beingarranged-to radiate from acommon center, but saidsheet or-plate being cut out intirevicinity-of-said common center to permit of insertion of said movable contact member, and the latter being adapted-to=rotate in a circular path in said cut-out ,portionand to contact a 'fixed number of thegirnprintedlines inany of its :positions, while said fixed contact member'is locatedalong the periphery of said assembly andis shapedto outline With'its inner edge a preassignedmathematical curvewith respect to the degree of;ro tation ofsaid movable contact member. a

7. An electricalresistor as incl aim l, the imprinted lines being arranged to radiate from a common center, but saidsheet oriplatebeing cut out in the vicinityof said common center to'permit insertion of said'fixed contact member; said sheet or plate being of circular outline along its periphery and said movable contact member being adaptedttorotatealong said circular-periphery and towntact -..a fixed .number of the imprinted lines in any rof;its positions,;.while. said,internallysituated fixed contact memher is shaped to: outline ,with its outer edgeapreassigned mathematical curve with respect to the degree ofrotation of said movable contact member.

8. A variable electricalresistor comprising a support having anon-conductive surface of substantial area, an assembly of adjacent, discrete, conductive linear segments laid oifonsaid surfacetheextent of said assembly in the dimension'crosswise of said linear segments being .substantialcompared to the average length of ,said segments, said conductive linear segments-being of substantial and uniform resistivity flengthwise, and being relatively thin and uniformlyspacedjfrom each other at relatively narrow-intervals compared to the crosswise extent of said assembly wherebyto present the appearance of a fine grating on' the face ofsaid support, the lengths of said conductive segments-varying continuously from one edge of the assembly to the other and being defined by a fixed conducting element "traversing all said conductive segments on one end thereof and by a predetermined-line of contact with a' moving contactor on the other end thereof, said-fixed conductin g elementbeingin electrical contact with each of said'linear segments wherebyto connect'said segments in'electrically parallel connection at one end thereof and-having-its 'edge which facesinwardly toward said grating shaped in the form of apreassigned mathematical-curve Whose-ordinatesvary in a predetermined manner-with respect to the abscissa thereof, whereby the totalresistance of eachconductive linear segment is determinedby the-length thereof included between the path of said moving contactor and the-inner edge of saidfixed conducting element, and wherebythe net resistance of any group of saidconductingsegments contacted simultaneously by a moving contactor along said predetermined path will .vary with theposition of said moving contactor according to'a predetermined mathematical formula.

9. A variable resistoras defined in claim 8 in combination withJa, movable contactor adapted to move at a constant angle with respect'to said conductive segments and to contacta-limited number of said segments simultaneously, whereby the group of conductive segments contactedby it as it moves along said predetermined path varies in identity but remains constant in number, and wherebythe 'variationin the net resistance of the group of conductingsegments contacted follows a mathematical curve similar tothe curve defining the inner edge of said fixed conducting-element.

10. A variable resistor as defined in claim 8 in combinationwith amovable contactoradapted to move at a constant anglewith respectto saidconductive segments and to-contact-simultaneously all-the conductive-linear segments included-between one edge of said assembly and the position of the foremost; point on the contacting area of said moving contactoi'gwherebythe net resistance of the group ofconducting elements simultaneously contaetedis determined by'the mathematical inverse-0f the summarized conductance thereof, and whereby said net resistance varies along a predetermined mathematical curve which-is dependent upon-but different from the curve defining the inner edge ofsaid fixed conducting element.

i=1. As an element of an electric circuit, a body of dielectric material having a surface of substantial width compared to its length, a cornb like assembly-of adjacent, discrete, linear semiconductors covering by their number essentially the entire width of said surface, said linear semi-conductors being of uniform resistivity lengthwise and of substantial resistanceperuni-t length, and a conductorbf substantial-area extending along-the width of saidsurface and beingincontact with said linear semiconductors v.wherebyto connect all of them in parallel, said conductorhaving itstinneredge shapedtin the form of a ,selectedmathematical curve at 'least aportion of which is nonlinear,,andsaidsemiconductors being disconnected from each other at the ends opposite their contact with said transverse conductor, and having said ends aligned ,for successi-ve contact with a movable contactor which moves transversely of said semiconductors at a distance from said curved inner edge, whereby the segment 17 of each semiconductor included between the inner edge and the path of said movable contactor will have a resistance value varying in mathematical correspondence to the function defined by said inner edge with respect to said path.

12. A variable resistor comprising a non-conducting, plastic sheet having imprinted thereon an assembly of relatively conducting elements of uniform width and thickness in closely spaced, parallel relation to each other to outline a pattern of limited area but of substantial width compared to its average height, one edge of said sheet being trimmed to give the pattern an essentially straight-line border cutting across said parallel conducting elements, and the lengths of the imprinted lines as measured from said straight edge being delimited progressively by a continuous, electrically conducting strip to outline a curve bearing a predetermined mathematical relation to an X-axis in the vicinity of said straight edge, said sheet being supported on a shaped, rigid body adapted to give the assembly a permanent form, and movable contact means associated with said straight edge and adapted to connect in electrically parallel relation a selected group of said conducting elements, whereby electrical current may be sent through said movable contact means into and through said selected group of conducting elements into said fixed contacting means and lengthwise of the latter to a selected point thereon, said conducting elements being of sufiiciently high resistance to render the resistance of said electrically conducting strip and said movable contact means negligible by comparison, and having a sufiiciently high resistivity per unit length to render the resistance of said selected group material with respect to other electrical elements intended to be used in series circuit therewith.

13. A variable resistor as in claim 12, said sheet being supported on a rigid body of cylindrical form, said imprinted conducting elements being disposed lengthwise of said cylindrical body, said straight edge of said sheet 18 being disposed in the form of an incomplete circle along the base of said cylindrical body, and said movable contact means being adapted to rotate along the base of the cylinder and to contact, in any position thereof, a fixed number of said imprinted elements along said circular edge.

14. A variable resistor as in claim 12, the straight edge of the imprinted sheet being formed by bending a portion of the imprinted sheet over a straight edge of the supporting rigid body, and said movable contact means being arranged to make contact with said bent over portion against the edge of said supporting body.

15. A variable resistor as in claim 12, the device comprising further a flexible ribbon of conducting material interposed between said moving contact means and the imprinted sheet along said straight edge, whereby to eliminate sliding friction between said moving contact means and said imprinted sheet.

References Cited in the file of this patent UNlTED STATES PATENTS Re. 23,219 Moore Apr. 11, 1950 437,111 Gunning Sept. 23, 1890 1,872,954 Hunkins Apr. 23, 1932 1,940,102 Robertson Dec. 19, 1933 2,119,195 Bagno May 31, 1938 2,381,733 Fischer Aug. 7, 1945 2,457,178 Richardson Dec. 28, 1948 2,513,415 Larsen July 4, 1950 2,529,123 Arnold Nov. 7, 1950 2,597,674 Robbins May 20, 1952 2,616,994 Luhn Nov. 4, 1952 2,653,206 Montgomery Sept. 22, 1953 2,759,078 Brown Aug. 14, 1956 FOREIGN PATENTS 675,392 Great Britain July 9, 1952 UNITED STATES PATENT OFFICE Certificate of Correction Patent No. 2,833,901 May 6, 1958 -David Katz It is hereb certified that error appears in the printed specification of the above num ered patent requiring correction and that the said Letters Patent should read as corrected below.

golumns 9 and 10, Table 2, third column thereof, third item from the top, or

a a read Sin I... 1 1 Sin lL Z 1" colunm 11, lines 47 to 49, for

(1' L\ )=f read F(X) =f column 13, line 15, for flop read -flap; line 71, for continguously read contigu0usly-; column 17, line 8, after relatively insert thin-.

Signed and sealed this 5th day of August 1958.

Attest: KARL H. AXLINE, ROBERT C. WATSON, Attesting Oficer. Gammissz'oner of Patents.

UNITED STATES PATENT OFFICE Certificate of Correction Patent No. 2,833,901 May 6, 1958 David Katz It is hereby certified that error appears in the printed specification of the above numbered patent requiring correction and that the said Letters Patent should read as corrected below.

Columns 9 and 10, Table 2, third column thereof, third item from the top, for

a read a sin -8 sin --xsin xi- 2 2 2 2 colunm 11, lines 47 to 49, for

(FX)=I% read F(X) =1 column 13, line 15, for flop read fla line 71, for continguously read contiguous1y; column 17, line 8, after relatively insert thin.

Signed and sealed this 5th day of August 1958.

Attest: KARL H. AXLINE, ROBERT C. WATSON, Attesting Ofiiaer. Oommz'ssz'oner 'of Patents. 

